If m and n are natural numbers such that n 1, and m = 2 25 × 3 40 , then m - n equals

Question

If m and n are natural numbers such that n > 1, and m = 2 25 &times; 3 40 , then m - n equals

Accepted Answer

209947 — Medium The intended relation is $m^{\,n}=2^{25}	imes3^{40}$ with $n>1$ a natural number. For $m$ to be a natural number, $n$ must divide both exponents, so take $n=\gcd(25,40)$ to keep the bases as small powers; then compute $m-n$. 1 Require $n$ to divide the exponents. If $m^{\,n}=2^{25}\cdot3^{40}$ and $m$ is a natural number, then $m=2^{25/n}\cdot 3^{40/n}$, so $n\mid 25$ and $n\mid 40$. $$\begin{aligned} &n\mid\gcd(25,40)=5 \quad	ext{(common divisor of both exponents)}\ &\Rightarrow\ n=5

CAT 2024 Slot 2 — QA Question 8

Answer & solution